Type I and Type II Error Rates and Statistical Power Tradeoff

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Hypothesis Testing.

Your Task
For a simple null hypothesis $H_0: \theta = \theta_0$ versus simple alternative $H_1: \theta = \theta_1$, state the Neyman-Pearson Lemma. Prove that the Likelihood Ratio Test (LRT) maximizes statistical power $1 - \beta$ for any fixed significance level $\alpha$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.